Receive updates to calendars on iCloud.com

You can receive an email message any time there’s an update to a shared calendar you own or participate in.

Go to icloud.com/calendar, then sign in to your Apple Account (if necessary).
Click at the top of the sidebar, then choose Settings.
In the Events pane, select “Email me shared calendar updates.”
To stop receiving updates, deselect the same option.
Click Save.

If you own the shared calendar, the email is sent to the address you set up to send invitations. See Invite people to a calendar event on iCloud.com. If you’ve been invited to the shared calendar, the email is sent to the address used to invite you to the calendar.

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iCloud User Guide

Receive updates to calendars on iCloud.com